Abstract

In the study by Papanastassiou and Papachristodoulos, 2009 the notion of 𝑝-convergence in measure was introduced. In a natural way 𝑝-convergence in measure induces an equivalence relation
on the space 𝑀 of all sequences of measurable functions converging in
measure to zero. We show that the quotient space ℳ is a complete but
not compact metric space.

1. Introduction

Convergence in measure plays a fundamental role in several branches of Mathematics, for example in integration theory and in stochastic processes. In [1] a “Bochner-type” integration theory was developed in the context of Riesz spaces with respect to a convergence introduced axiomatically, and in particular some Vitali convergence theorems and Lebesgue dominated convergence theorems were proved. Similar subjects were investigated by Haluška and Hutník [2, 3] in the setting of operator theory for Bochner- and Dobrakov-type integrals (see also [4, 5]) and in [6–8] for the Kurzweil-Henstock integral in Riesz spaces.

In several contexts of integration theory it could be advisable to extend the concept of convergence in measure in order to get applications, for example, in the study of the stochastic integral and stochastic differential equations (see, e.g., [9]).

This paper is a continuation of [10], where the notion of 𝑝-convergence in measure was introduced. In this paper we investigate a structure related to the vector space 𝑀 of all converging sequences of measurable functions.

Let (Γ,Σ,𝜇) be an arbitrary measure space, where 𝜇 is a [0,∞]-valued measure, and let 𝑓𝑛,𝑓∶Γ→ℝ,𝑛=1,2,…, be measurable functions.

We adopt the following usual terminology. By the notation 𝑓𝑛𝜇→𝑓 we denote that the sequence of measurable functions (𝑓𝑛)𝑛 converges in measure to 𝑓. Also for a pair ((𝑓𝑛)𝑛,𝑓) and 𝜀≥0 we set 𝐴𝜀𝑛=||𝑓𝛾∈Γ∶𝑛||=||𝑓(𝛾)−𝑓(𝛾)≥𝜀𝑛||−𝑓≥𝜀,𝑛=1,2,….(1.1)
We denote by ℕ the set of all positive integers and 𝑐+0 the set of all real-valued nonnegative sequences (𝜀𝑛)𝑛 converging to 0. Also for 𝑝>0 we set ℓ+𝑝=𝜀𝑛𝑛∶𝜀𝑛≥0for𝑛=1,2,…,∞𝑛=1𝜀𝑝𝑛.<∞(1.2)
Convergence in measure is characterized by elements (𝜀𝑛)𝑛 of 𝑐+0 as follows: 𝑓𝑛𝜇⟶𝑓iﬀthereexists𝜀𝑛𝑛∈𝑐+0suchthatlim𝑛→∞𝜇𝐴𝜀𝑛𝑛=0.(1.3)
Taking into account that the sequence (𝜀𝑛)𝑛 above expresses the quality of approximation of (𝑓𝑛)𝑛 to 𝑓, in [10] the authors introduced the following notion of convergence which we call 𝑝-convergence in measure.

More precisely we say that, given 𝑝>0, (𝑓𝑛)𝑛𝑝-converges in measure to 𝑓 (and we write 𝑓𝑛𝑝−𝜇−−−→𝑓) if and only if there exists an element (𝜀𝑛)𝑛∈ℓ+𝑝 such that lim𝑛→∞𝜇𝐴𝜀𝑛𝑛=0.(1.4)
Obviously 𝑝-convergence in measure implies convergence in measure. It is proved (see [10, Preposition 2.3]) that if the measure 𝜇 is not trivial and 0<𝑝<𝑞, then 𝑝-convergence in measure implies 𝑞-convergence in measure, while the converse implication in general fails. So 𝑝-convergence in measure is strictly stronger than convergence in measure. As a consequence of the above result we have that 𝑀0≨𝑀𝑝≨𝑀𝑞≨𝑀∞≨𝑀,0<𝑝<𝑞,(1.5)
where𝑓𝑀=𝑛𝑛∶𝑓𝑛𝜇,𝑀⟶0𝑝=𝑓𝑛𝑛∶𝑓𝑛𝑝−𝜇𝑀−−−→0,𝑝>0,0=𝑝>0𝑀𝑝,𝑀∞=𝑝>0𝑀𝑝.(1.6)
We note that 𝑀 is considered as a vector space under usual operations and the notation 𝑁≨𝑀 means that 𝑁 is a proper vector space of 𝑀.

2. Metric Spaces of Sequences of Measurable Functions

In a natural way 𝑝-convergence in measure induces an equivalence relation on the vector space 𝑀={(𝑓𝑛)𝑛∶𝑓𝑛𝜇→0}. We consider 𝑀 as a subspace of 𝐿0(Γ)ℕ, ℵ0 copies of the vector space 𝐿0(Γ) of all real-valued measurable functions with the usual operations.

Definition 2.1. Let (𝑓𝑛)𝑛,(𝑔𝑛)𝑛 be elements of 𝑀. We say that (𝑓𝑛)𝑛, (𝑔𝑛)𝑛 are equivalent ((𝑓𝑛)𝑛∼(𝑔𝑛)𝑛) if and only if for each positive real number 𝑝 there exists an element (𝜀𝑛)𝑛 of ℓ+𝑝 such that
lim𝑛→∞𝜇||𝑓𝑛−𝑔𝑛||>𝜀𝑛=0(2.1)
or equivalently
𝑓𝑛−𝑔𝑛𝑝−𝜇𝑓−−−→0,∀𝑝>0⟺𝑛−𝑔𝑛𝑛∈𝑀0.(2.2)Since 𝑀0 is a vector subspace of 𝑀 the relation ~ is an equivalence one. We set ℳ=𝑀∼=𝑀/𝑀0.In the sequel we will define a metric 𝑑 on ℳ under which ℳ turns to be a complete metric space, similarly as a Fréchet space.

Definition 2.2. Let (𝑓𝑛)𝑛∈ℳ. We define
‖‖(𝑓𝑛)𝑛‖‖𝑓=arctaninf𝐴𝑛,(2.3)
where
𝐴𝑓𝑛=𝑝>0∶𝑓𝑛𝑝−𝜇=𝑓−−−→0𝑝>0∶𝑛𝑛∈𝑀𝑝.(2.4)By (1.5) it follows that 𝐴(𝑓𝑛) is an interval in ℝ.

Remarks 2.3. (i) We note that the above set 𝐴(𝑓𝑛) could be empty. In this case we set ‖(𝑓𝑛)𝑛‖=𝜋/2.(ii) If (𝑓𝑛)𝑛∼(𝑔𝑛)𝑛, then ‖(𝑓𝑛)𝑛‖=‖(𝑔𝑛)𝑛‖. (Indeed, as (𝑔𝑛)𝑛=(𝑔𝑛−𝑓𝑛)𝑛+(𝑓𝑛)𝑛 and (𝑓𝑛)𝑛∈𝑀𝑝, it follows that (𝑓𝑛)𝑛∈𝑀𝑝 if and only if (𝑔𝑛)𝑛∈𝑀𝑝.)(iii) We set ℳ𝑝=𝑀𝑝/∼,𝑝>0, and hence ℳ𝑝 is a proper vector subspace of ℳ and the following strict inclusion holds:
ℳ𝑝1≨ℳ𝑝2if0<𝑝1<𝑝2(see(1.5)).(2.5)

Hence, ℳ becomes a metric space and the metric 𝑑((𝑓𝑛)𝑛,(𝑔𝑛)𝑛)=‖(𝑓𝑛−𝑔𝑛)𝑛‖ is invariant under translations.

Proof. (i) It is obvious.(ii) If (𝑓𝑛)𝑛∼(0)𝑛, then ‖(𝑓𝑛)𝑛‖=0.Conversely, if ‖(𝑓𝑛)𝑛‖=0, then, (𝑓𝑛)𝑛𝑝−𝜇−−−→0 for each 𝑝>0, and hence (𝑓𝑛)𝑛∈ℳ0 and consequently (𝑓𝑛)∼(0)𝑛.(iii) For 𝑎≠0, it holds that
𝐴𝜀𝑛𝑛=||𝑓𝑛||≥𝜀𝑛=||𝑎𝑓𝑛||≥|𝑎|𝜀𝑛,∞𝑛=1𝜀𝑝𝑛<∞⟺∞𝑛=1|𝑎|𝜀𝑛𝑝<∞,(2.6)
for each sequence (𝜀𝑛)𝑛 of positive real numbers. Hence,
𝑓𝑛𝑛𝑝−𝜇−−−→0iﬀ𝑎𝑓𝑛𝑛𝑝−𝜇−−−→0,(2.7)
which means that ‖(𝑓𝑛)𝑛‖=‖(𝑎𝑓𝑛)𝑛‖.(iv) The inequality is obvious if ‖(𝑓𝑛)𝑛‖=𝜋/2 or ‖(𝑔𝑛)𝑛‖=𝜋/2.Suppose ‖(𝑓𝑛)𝑛‖≤‖(𝑔𝑛)𝑛‖<𝜋/2. Then we conclude that 𝐴(𝑔𝑛)⊂𝐴(𝑓𝑛). Hence, (𝑔𝑛)𝑛𝑝−𝜇−−−→0 implies (𝑓𝑛)𝑛+(𝑔𝑛)𝑛𝑝−𝜇−−−→0. So 𝐴(𝑔𝑛)⊆𝐴(𝑓𝑛+𝑔𝑛), which implies that
‖‖𝑓𝑛+𝑔𝑛𝑛‖‖≤‖‖𝑔𝑛𝑛‖‖≤‖‖𝑓𝑛𝑛‖‖+‖‖𝑔𝑛𝑛‖‖.(2.8)

Theorem 2.5. The space (ℳ,𝑑) is a complete metric space.

Proof. Let (𝐹𝑛)𝑛 be a Cauchy sequence in ℳ, where 𝐹𝑛=(𝑓𝑛,𝑖)𝑖,𝑛=1,2,…. Hence, there exists an increasing sequence of positive integers (𝑛𝑘)𝑘 such that
‖‖𝐹𝑛−𝐹𝑚‖‖1<arctan𝑘,for𝑛,𝑚≥𝑛𝑘,𝑘=1,2,….(2.9)
This means that, for each 𝑛,𝑚≥𝑛𝑘, there exists a sequence (𝜀𝑛,𝑚,𝑖)𝑖 of positive real numbers with ∑∞𝑖=1𝜀1/𝑘𝑛,𝑚,𝑖<∞ such that
𝜇𝐴𝑛,𝑚,𝑖⟶0,𝑖⟶∞,where𝐴𝑛,𝑚,𝑖=||𝑓𝑛,𝑖−𝑓𝑚,𝑖||≥𝜀𝑛,𝑚,𝑖.(2.10)
Then,
∞𝑖=1max𝑛ℓ≤𝑛≤𝑛𝑘+1𝜀𝑛,𝑛𝑘+1,𝑖1/ℓ<∞,forℓ=1,2,…,𝑘,𝑘∈ℕ.(2.11)
From (2.10) and (2.11), proceeding by induction, it follows that there exists an increasing sequence (𝑖𝑘)𝑘 of positive integers such that for each 𝑘 we have
𝜇𝐴𝑛,𝑛𝑘+1,𝑖<1𝑘for𝑖≥𝑖𝑘,𝑛1≤𝑛≤𝑛𝑘+1,𝑘=1,2,…,(2.12)∞𝑖=𝑖𝑘max𝑛ℓ≤𝑛≤𝑛𝑘+1𝜀𝑛,𝑚,𝑖1/ℓ<12𝑘,forℓ=1,2,…,𝑘,(2.13)𝜇||𝑓𝑛𝑘+1,𝑖||≥1𝑘<1𝑘,for𝑖≥𝑖𝑘,(2.14)which express the uniform convergence to zero of finite number of sequence which converges to zero and a finite number of tails of convergence series.We set
𝑓𝐹=𝑛1,1,𝑓𝑛1,2,…,𝑓𝑛1,𝑖1−1;𝑓𝑛2,𝑖1,…,𝑓𝑛2,𝑖2−1;…;𝑓𝑛𝑘+1,𝑖𝑘,…,𝑓𝑛𝑘+1,𝑖𝑘+1−1=𝑓;…𝑖𝑖.(2.15)
By (2.14) it follows that 𝐹=(𝑓𝑖)𝑖∈ℳ. We have to show that
‖‖𝐹𝑛‖‖−𝐹⟶0,𝑛⟶∞⟺∀ℓ∈ℕ∃𝑛0𝑓∈ℕ∶𝑛,𝑖−𝑓𝑖𝑖(1/ℓ)−𝜇⟶0,for𝑛≥𝑛0.(2.16)
This means that we have to find 𝑛0∈ℕ and for 𝑛≥𝑛0 a sequence of positive real numbers (𝜀𝑖)𝑖 with ∑∞𝑖=1𝜀𝑖1/ℓ<∞ such that
𝜇𝐴𝑖⟶0,𝑖⟶∞,where𝐴𝑖=||𝑓𝑛,𝑖−𝑓𝑖||≥𝜀𝑖.(2.17)
Indeed let ℓ∈ℕ,𝑛≥𝑛0=𝑛ℓ, and 𝑛ℓ≤𝑛𝑘<𝑛<𝑛𝑘+1 for some 𝑘∈ℕ.We set
𝜀𝑖=1,if𝑖=1,2,…,𝑖𝑘𝜀−1,𝑖=𝜀𝑛,𝑛𝑘+1,𝑖,if𝑖=𝑖𝑘,…,𝑖𝑘+1𝜀−1,𝑖=𝜀𝑛,𝑛𝑘+2,𝑖,if𝑖=𝑖𝑘+1,…,𝑖𝑘+2−1,(2.18)
and so on.It holds that
∞𝑖=1𝜀𝑖1/ℓ≤𝑖𝑘+1−12𝑘+12𝑘+1+⋯<∞(by(2.13)).(2.19)
Also, for 𝑖𝑚≤𝑖≤𝑖𝑚+1,𝑚≥𝑘, we have that
𝐴𝑖=||𝑓𝑛,𝑖−𝑓𝑖||≥𝜀𝑖=||𝑓𝑛,𝑖−𝑓𝑛𝑚+1,𝑖||≥𝜀𝑛,𝑛𝑚+1,𝑖(2.20)
(by definition of 𝑓𝑖). Hence, by (2.12), we take
𝜇𝐴𝑖<1𝑚.(2.21)
This implies (2.17), and the proof is complete.

Remarks 2.6. It is easy to see the following (a)The addition +∶((𝑓𝑛)𝑛,(𝑔𝑛)𝑛)↦(𝑓𝑛+𝑔𝑛)𝑛 is continuous.(b)The translation 𝑇(𝑔𝑛)𝑛∶(𝑓𝑛)𝑛↦(𝑓𝑛)𝑛+(𝑔𝑛)𝑛=(𝑓𝑛+𝑔𝑛)𝑛 is a homeomorphism. Hence, the system of neighborhoods of (0)𝑛 determines the topology of (ℳ,𝑑).(c)The multiplication operator
𝐻𝑎∶𝑓𝑛𝑛𝑓⟼𝑎⋅𝑛𝑛,𝑎≠0,(2.22)
is a homeomorphism.(d)The multiplication (𝑎,(𝑓𝑛)𝑛)↦𝑎(𝑓𝑛)𝑛 is not continuous. (If 𝑎𝑛→0,𝑎𝑛≠0,𝑛=1,2,…, and 𝐹=(𝑓𝑛)𝑛∈ℳ, 𝐹≠0, and 𝐹𝑛=𝐹 for 𝑛=1,2,…, it holds that
𝑎𝑛⟶𝑎≠0,𝐹𝑛𝑑⟶𝐹,(2.23)
but ‖𝑎𝑛𝐹𝑛−0𝐹‖=‖𝑎𝑛𝐹𝑛‖=‖𝐹𝑛‖=‖𝐹‖↛0).(e)The family (ℳ𝑝)𝑝>0 is a system of neighborhoods of (0)𝑛. (Indeed, if 𝑆𝑟=𝑆((0)𝑛,𝑟)={(𝑓𝑛)𝑛∶‖(𝑓𝑛)𝑛‖<𝑟} for 𝑟>0, then for 0<𝑟1<𝑝<𝑟2 we have 𝑆𝑟1⊂ℳ𝑝⊂𝑆𝑟2.)Though (ℳ,𝑑) is not a topological vector space, (ℳ,𝑑) is complete and the subspaces ℳ𝑝, 𝑝>0 constitute a system of closed and convex neighborhoods of (0)𝑛, as we will see in the sequel (Proposition 2.7). Hence, (ℳ,𝑑) is something like a Fréchet space. For example, the principle of uniform boundedness holds true, as for this principle only continuity of 𝐻𝑎 is needed (see [11]).

Proposition 2.7. The subspaces ℳ𝑝 are closed for each 𝑝>0.

Proof. Suppose that 𝑝>0 and (𝐹𝑛)𝑛 is a sequence in ℳ𝑝, where 𝐹𝑛=(𝑓𝑛,𝑖)𝑖, 𝑛=1,2,…, and 𝐹=(𝑓𝑛)𝑛∈ℳ such that
𝐹𝑛𝑑‖‖𝐹⟶𝐹⟺𝑛‖‖−𝐹⟶0,𝑛→∞.(2.24)
Hence, there exist 𝑝′<𝑝 and 𝑛0 such that
𝐹𝑛0−𝐹𝑝−𝜇−−−→0.(2.25)
This implies that 𝐹𝑛0−𝐹∈ℳ𝑝⊂ℳ𝑝, and, since 𝐹𝑛0∈ℳ𝑝, it follows that 𝐹∈ℳ𝑝.

Proposition 2.8. ℳ∞=⋃𝑝>0ℳ𝑝 is a closed subspace of ℳ.

Proof. Suppose 𝐹0=(𝑓0,𝑖)𝑖∉ℳ∞, then 𝐹0+ℳ𝑞,𝑞>0 is a neighborhood of 𝐹0 and (𝐹0+ℳ𝑞)∩ℳ𝑝=∅ for all 𝑝>0.Indeed, if 𝐹0+𝐹1=𝐹2 for some 𝐹1∈ℳ𝑞 and some 𝐹2∈ℳ𝑝, then 𝐹0∈ℳ𝑟, where 𝑟=max(𝑝,𝑞), which is a contradiction. Hence, (𝐹0+ℳ𝑞)∩ℳ∞=∅, which implies that ℳ∞ is closed.

Remark 2.9. If S((0)𝑛,𝑟) denotes the open sphere with center (0)𝑛 and radius 𝑟, it is easy to see that the family𝑆(0)𝑛,𝑟𝑟>0∪𝐹+𝑆(0)𝑛,𝑟𝐹∉ℳ∞,𝑟>0(2.26)
is an open covering of ℳ without a finite subcovering. Hence ℳ is not compact.

Acknowledgment

N. Papanastassiou work is supported by the Universities of Perugia and Athens.